I am 40 years old, and I have never been drunk. I live my life like if someone does not like that I abstain from alcohol, then they are not really a good friend for me. This is a personal choice. It is deeper than just a fear. It is a knowledge that I tried the social scene of drinking in my 20s and 30s--it did not make me a better person--and then I saw what it did to my husband. And he is a great guy--but he can be very hurtful when he drinks. I never imagined I would be married to an alcoholic. It really sent me running back to God. And I have been very thankful to God to be there for me during the most difficult times.
You cannot control your cousin. You can control yourself. You can choose to surround yourself with friends who respect your view on drinking. Take care of yourself first.
You are still young and trying to figure out who you want to be. Enjoy your youth. I would love to be 16 years again, and just enjoy life with my peers around that age. I just loved being involved with the youth groups at my church. We would go get pizza and soft drinks--there was just so much more innocence there. I would suggest more innocent things like meeting for pizza with your cousin or just meeting for ice cream or going to Starbucks or just going to the mall and going shopping.
Just do not let the drinking thing age you where you cannot enjoy your youth.